There are $3$ ways to specify a surface:
In $(1)$ we have a surface because we use one axis for $x$, one for $y$ and one for the output $z$.
In $(2)$ we have a surface because instead of having a four-dimensional graph, we do not consider the output to change, in fact we fix it to a constant value $c$ and plot for it, obtaining a level surface. So we use one axis for $x$, one for $y$ and one for $z$
In $(3)$ we have a surface because we use one axis for $s$, one for $t$ and one for the output.
However it seems to me that there are very different properties for these three options. For example, in $(1)$, in order to have a function, we need to have only one output coming from the inputs. However I think this is not always satisfied by the other two. Indeed sometimes there are sketches of $(3)$ that clearly violate that.
Reading this question helped me make more clear the situation in $2d$: Why is an ellipse, hyperbola, and circle not a function?
However I am not sure I understand the difference between these. For example, in $(3)$, do we always have to define the interval of the parameters in a closed interval? What if we defined it on the whole $\mathbb{R}^2$? And furthermore, do $(2)$ and $(3)$ respect the criterias to be functions? For example the one stated above, but one can think of others I guess.
So far I think that $(2)$ is not a function when we take multiple values of $c$, but it is a function when we plot one and only one value of $c$.Is this correct?
Concerning $(3)$ this is often thought of as a path when we have only one parameter (for example $t$ is time), however one could consider this as having two parameters as position and time maybe. Is this curve necessarily a function? What does it have in common with a function?
so summarizing:
Thank you
